First off, the officer is under no obligation to blue light someone immediately after observing a violation.
What we have here is someone sliding to a stop at a red light. I suppose you could articulate that as Too Fast for Conditions (OCGA 40-6-180), but I think that's a stretch. I think it is reasonable suspicion of someone who is having some driving issues. It points to someone who is driving too quickly or not paying close enough attention to what he's doing. That's why the officer turned around.
Then the driver drove on the left side of a road without a center lane marker. That's a probable cause to stop someone for OCGA 40-6-40. Georgia’s weaving statute (OCGA 40-6-48) only refers to roads laned for traffic. However, weaving could still be articulatable suspicion that would justify a traffic stop.
At this point the officer has cause to stop the person, but he doesn't. I think the Georgia courts would see that as irrelevant. There are plenty of appellate decisions about officers following DUI suspects into neighboring jurisdictions, much less a short distance. (What was it, .3 miles?)
Then the officer goes onto privately maintained roads. In Georgia, an officer, or anyone else, can go onto another's property unless it's posted against trespassing. Those neighborhoods that are gated generally want police presence, but that's not really a legal issue.
Since the officer has probable cause of a misdemeanor committed in his presence, the officer is legally in "hot pursuit," and he can follow you wherever you go. Legally, he doesn't have to turn on his lights and siren for it to be "hot pursuit" under the law.
Now he finally decides to stop you once you reach your personal property. That's fine. If he had made no attempt to stop you before you entered your home, we'd be in a different situation. You can't break into a person's home to arrest them for a misdemeanor without a warrant, some very serious exigent circumstances, or actual pursuit (a foot chase).
All of the actions you describe sound legal. You don't even mention anything that sounds unreasonable.
A note on private property traffic enforcement in Georgia: DUI in the state of Georgia applies to any location in the state. I arrested a guy on a four wheeler in his backyard for DUI. Most moving violations are not a crime in Georgia if they're on private property, unless it is a commonly used cut through to get from one road to another or the owner of the private property requests traffic enforcement (which is almost never done).
Since DUI, can be enforced on private property, this whole incident could have taken place in a gated community, because the officer would have had plenty of articulatable suspicion of DUI to make a stop.